We consider the shear flow behavior of nematic LCPs, modeled via an extension of the Doi theory that incorporates the mean-field nematic potential due to Marrucci and Greco to account for distortional elasticity. Based upon the constitutive model that derives from this starting point, we utilize finite-element methods to investigate the LCP behavior in a planar shear flow. We assume that the LCP is pinned at the walls and is initially in its equilibrium configuration. The goal of our simulations is to explore the evolution of the LCP structure and the flow. Our results show that in-plane tumbling instabilities lead to a non-uniform orientation field, which, in turn, arrests tumbling. The resulting quasi-steady-state texture is characterized by a length scale that seems to be consistent with a Marrucei-like scaling. When we allow for out-of-plane tipping of the director, we predict an out-of-plane director instability, which is qualitatively consistent with what has been observed in experiments.